Global number field 40.40.32473210254684090614318847656250000000000000000000000000000000000000000.1: \(\Q(\zeta_{300})^+\)

• Label: 40.40.3247321025...0000.1
• Degree: $40$
• Signature: $[40, 0]$
• Discriminant: $2^{40}\cdot 3^{20}\cdot 5^{70}$
• Root discriminant: $57.91$
• Ramified primes: $2, 3, 5$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} - 40 x^{38} + 740 x^{36} - 8400 x^{34} + 65450 x^{32} - 371007 x^{30} + 1582210 x^{28} - 5177835 x^{26} + 13144625 x^{24} - 25995750 x^{22} + 39996264 x^{20} - 47552155 x^{18} + 43139880 x^{16} - 29279950 x^{14} + 14438100 x^{12} - 4952883 x^{10} + 1106455 x^{8} - 144655 x^{6} + 9150 x^{4} - 200 x^{2} + 1 \)

Invariants

Degree:		$40$
Signature:		$[40, 0]$
Discriminant:		\(32473210254684090614318847656250000000000000000000000000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{70}\)
Root discriminant:		$57.91$
Ramified primes:		$2, 3, 5$
This field is Galois and abelian over $\Q$.
Conductor:		\(300=2^{2}\cdot 3\cdot 5^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(131,·)$, $\chi_{300}(7,·)$, $\chi_{300}(137,·)$, $\chi_{300}(11,·)$, $\chi_{300}(17,·)$, $\chi_{300}(283,·)$, $\chi_{300}(289,·)$, $\chi_{300}(191,·)$, $\chi_{300}(163,·)$, $\chi_{300}(293,·)$, $\chi_{300}(113,·)$, $\chi_{300}(169,·)$, $\chi_{300}(299,·)$, $\chi_{300}(257,·)$, $\chi_{300}(173,·)$, $\chi_{300}(49,·)$, $\chi_{300}(179,·)$, $\chi_{300}(181,·)$, $\chi_{300}(59,·)$, $\chi_{300}(61,·)$, $\chi_{300}(53,·)$, $\chi_{300}(67,·)$, $\chi_{300}(197,·)$, $\chi_{300}(71,·)$, $\chi_{300}(119,·)$, $\chi_{300}(77,·)$, $\chi_{300}(43,·)$, $\chi_{300}(223,·)$, $\chi_{300}(187,·)$, $\chi_{300}(229,·)$, $\chi_{300}(103,·)$, $\chi_{300}(233,·)$, $\chi_{300}(109,·)$, $\chi_{300}(239,·)$, $\chi_{300}(241,·)$, $\chi_{300}(247,·)$, $\chi_{300}(121,·)$, $\chi_{300}(251,·)$, $\chi_{300}(127,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$39$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 31857175264724490000000 \) (assuming GRH)

Galois group

$C_2\times C_{20}$ (as 40T2):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2\times C_{20}$

Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), 5.5.390625.1, \(\Q(\zeta_{60})^+\), 10.10.37968750000000000.1, \(\Q(\zeta_{25})^+\), 10.10.189843750000000000.1, 20.20.36040649414062500000000000000000000.1, \(\Q(\zeta_{100})^+\), \(\Q(\zeta_{75})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
2	Data not computed
3	Data not computed
5	Data not computed